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Second-Order Optimality Conditions for Nonconvex Set-Constrained Optimization Problems

Author

Listed:
  • Helmut Gfrerer

    (Institute of Computational Mathematics, Johannes Kepler University Linz, A-4040 Linz, Austria)

  • Jane J. Ye

    (Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada)

  • Jinchuan Zhou

    (Department of Statistics, School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, P.R. China)

Abstract

In this paper, we study second-order optimality conditions for nonconvex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper, we propose two approaches for establishing second-order optimality conditions for the nonconvex case. In the first approach, we extend the concept of the support function so that it is applicable to general nonconvex set-constrained problems, whereas in the second approach, we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

Suggested Citation

  • Helmut Gfrerer & Jane J. Ye & Jinchuan Zhou, 2022. "Second-Order Optimality Conditions for Nonconvex Set-Constrained Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 47(3), pages 2344-2365, August.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:3:p:2344-2365
    DOI: 10.1287/moor.2021.1211
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    References listed on IDEAS

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    1. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
    2. J. J. Ye & X. Y. Ye, 1997. "Necessary Optimality Conditions for Optimization Problems with Variational Inequality Constraints," Mathematics of Operations Research, INFORMS, vol. 22(4), pages 977-997, November.
    3. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2013. "Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 33-64, July.
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    Cited by:

    1. Xiaoxiao Ma & Wei Ouyang & Jane J. Ye & Binbin Zhang, 2025. "On second-order weak sharp minima of general nonconvex set-constrained optimization problems," Journal of Optimization Theory and Applications, Springer, vol. 207(2), pages 1-24, November.
    2. Jiawei Chen & Luyu Liu & Yuhong Dai & Elisabeth Köbis, 2026. "Directionally Variational Analysis and Second-Order Optimality Conditions for Mathematical Programs with Switching Constraints," Journal of Optimization Theory and Applications, Springer, vol. 208(1), pages 1-45, January.
    3. Ashkan Mohammadi & Ebrahim Sarabi, 2025. "Parabolic Regularity of Spectral Functions," Mathematics of Operations Research, INFORMS, vol. 50(3), pages 2017-2046, August.

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